The relevant legislation is
Taxes Consolidation Act 1997 s. 71.
Under subsection (1) the default rule is that the full amount of income arising from property/investments held outside the state is subject to income tax.
However sub (2) creates an exception; it says that sub (1) doesn't apply to someone who satisfied the Revenue Commissioners that they are not domiciled in the state.
Sub (3) sets out the alternative rule which is to apply in that case: tax is to be computed on the full amount of the actual sums received in the State:
- from remittances payable in the State, or
- from property imported into the State, or
- from money or value brought into the State arising from property not imported, or
- from money or value received in the State on credit or on account in respect of expected remittances, property, money or value brought into the State
Assume your foreign income is dividends paid on shares held abroad.
Item 1 will subject you to tax if you receive a cheque, transfer, etc sent from abroad representing some or all of your dividend income
Item 2 will subject you to tax if you spend your dividend income on property/goods of any kind, and then import that into the state.
Item 3 will subject you to tax if you use your dividends to buy property/goods abroad, do not import that, but then rent it out or sell it or otherwise derive money from it, and bring that money into the state. Note that this could happen years after you earned the income and bought the property; it's still a remittance which is taxable in the year that you bring it into the state.
Item 4 will subject you to tax if you receive borrowings in the state (could be from an Irish lender; could be from a foreign lender; doesn't matter) on the strength of your investment income/property that you haven't imported. So if you borrow from the bank, saying "I'll pay you back out of my overseas dividends/property when I'm no longer resident in Ireland" that doesn't work; the borrowing will be treated as a remittance. If you have a credit card issued by a foreign bank and you use it to pay for good/services received/consumed in Ireland, settling it out of your foreign dividend income — that's a remittance, and it's taxable.
But you avoid tax if you receive the value of the income abroad and keep it abroad, even if you consume it abroad. So if you leave Ireland for a holiday abroad and, while abroad, use your foreign dividend income to pay your hotel bills, restaurant bills, car hire, theatre tickets and enormous bar bill, that's fine; no liability to Irish income tax. (Don't use it to pay for souvenirs that you will bring back to Ireland, though.) You own a house in your home country and you use your dividend income to pay expenses associated with that house — rates, water bill, maintenance, occasional check by a caretaker; that's fine. Similarly you can use your foreign income e.g. to pay for services that you receive abroad. The Spanish lawyer you hired to smooth over that embarrassing little incident in Barcelona involving an exotic dancer and a goat? You can pay him, no problem.
Airline tickets between Ireland and abroad might be a bit of a gray area. I would say that if you use your foreign dividends to pay for a flight from Dublin to Paris, that's value brought into the state because the flight originates in Dublin and the cost is payable here. But if you book a flight from Paris to Dublin there's at least an argument that you haven't brought value into the State — by the time you land, the value of the flight has been entirely consumed and therefore you are not bringing anything in.